$p(X = c)=1$ then $E(X) = c$

Question

Let $X$ be an aleatoric number. If $X \equiv c$ then $E(X) = c$. But if $p(X = c)=1$ how can I show, starting from the axioms of expectation or easy properties (e.g. Chebyshev inequality), that $E(X) = c$?

Answer 1

This is more or less a direct computation of the expectation.

Note that $p (X^{-1} \{c\}^c) = 1- p (X^{-1} \{c \}) = 0$. Hence $ 1_{\{c\}^c}(\omega) X(\omega) = 0$ ae. [$p$].

$E X = \int X dp = \int (1_{\{c\}}X dp + 1_{\{c\}^c}X ) dp = \int 1_{\{c\}}c\, dp + \int 1_{\{c\}^c} X dp =c p (X^{-1} \{c \}) = c$